Colimits of internal categories

TL;DR

This paper investigates the existence and properties of colimits in the 2-category of internal categories within an extensive category, establishing conditions under which these colimits exist and are well-behaved.

Contribution

It provides new conditions ensuring that the 2-category of internal categories has finite 2-colimits and explores their stability and extensiveness properties.

Findings

The 2-category of internal categories has finite 2-colimits under certain conditions.

Codescent coequalisers are stable under pullback along discrete Conduché fibrations.

The paper establishes converse results related to these properties.

Abstract

We show that for an extensive $1$-category $\mathcal{E}$ with pullbacks and pullback stable coequalisers in which the forgetful functor $\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E})$ has left adjoint, the $2$-category $\mathbf{Cat}(\mathcal{E})$ of internal categories, functors and natural transformations has finite $2$-colimits. In addition, $\mathbf{Cat}(\mathcal{E})$ is extensive, has pullbacks and codescent coequalisers are stable under pullback along discrete Conduch\'{e} fibrations. Moreover, we give converse results to this.